Fortuin–Kasteleyn random-cluster unification extended to plaquette lattice gauge duality — E8 Intelligence Research

FINDING: Fortuin–Kasteleyn (FK) representation unifies percolation, Ising, and Potts models via a single random-cluster measure; Schweinhart extends this to plaquette (2-cell) variables, yielding a lattice gauge theory dual. MATH: - FK random-cluster measure on graph G=(V,E): Z_RC(q,v) = Σ_{A⊆E} q^{k(A)} v^{|A|}, v = e^J − 1 where k(A) = number of connected components (clusters) of (V,A). - Potts partition function identity: Z_Potts(q,β) = Z_RC(q, v), q ∈ ℤ⁺ (Fortuin–Kasteleyn 1972). - Plaquette generalization (Schweinhart): sum over subsets P of 2-cells (plaquettes) of a cell complex, weight q^{b(P)} v^{|P|}, where b(P) = rank of H₁ of the plaquette-subcomplex (first Betti number = number of independent cycles/holes). This is the natural higher-dimensional FK model. - Duality: on a d-dimensional lattice, the plaquette RC model on the primal lattice is dual to a Potts lattice gauge theory on the dual lattice — the gauge group is ℤ_q, and the duality exchanges (d−2)-cell disorder Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com